Kinematic analysis of lower-mobility parallel manipulators using efficient algebraic tools

Since several decades parallel manipulators have been in the center of interest in academic research as well as in industrial applications. In comparison to serial manipulators parallel manipulators have a number of advantages as for example better dynamical characteristics and a better load to weight ratio. They are used as flight simulators, car simulators or milling machines. Many applications in industry do not need the full motion capability in orientation and translation. This is the reason why recently parallel manipulators with restricted mobility, which is adapted to the prescribed task, came into the center of academic research. This fact becomes immediately evident because a manipulator with less motors has less masses to move and is therefore cheaper to manufacture. Well investigated are manipulators that perform planar, spherical or Schönflies motions although such manipulators also have unexpected behaviour, as has been shown recently. But also lower-mobility manipulators with coupled motion in orientation and translation might be of interest in special practical applications. The kinematic properties of lower-mobility parallel manipulators are complicated and not very well understood. But theoretical knowledge on the motion behaviour is an indispensable prerequisite for the design and the safe operation of the manipulator. This is the starting point of the project Kinematic analysis of lower-mobility parallel manipulators using efficient algebraic tools. In a pre-project financed by the Austrian-French Amadeus program we could show that methods from algebra and algebraic geometry and deep understanding of geometric features in higher dimensional parameter spaces can bring new and unexpected results concerning the kinematic properties of lower-mobility parallel manipulators. For the practical engineer interesting properties are: singularities, operation modes, transitions between operation modes, assembly modes, transition between assembly modes and parametric description of the work space. All mentioned properties can be derived from geometric-mathematical properties of algebraic varieties, derived from constraint equations which are linked to the design of the manipulator. In the project we will extend the findings of the pre-project to a full class of three degree of freedom parallel manipulators and develop new methods to deepen the understanding of the kinematic properties of these machines. this direction, but also broader by extending to other types of linkages.

The goal of this project was to obtain more knowledge on the global kinematic behavior of lower degree-of-freedom parallel manipulators with new and efficient algebraic tools. A lower mobility parallel manipulator has less than six degrees of freedom and usually exhibits different motion types called operation modes. In the scope of the project 3-RPS and 3-PRS parallel manipulators in all their different variants have been more specifically studied. These types of parallel robots have three degrees of freedom and consist of a mobile platform connected to a fixed base with three legs each consisting of a revolute joint (R), an actuated prismatic joint (P) and a spherical joint. The revolute and the spherical joints are passive. It was shown in the literature that both robot types are paradigmatic devices in the class of three-degree-of-freedom parallel manipulators and have very interesting kinematic features. We were especially interested in a complete description of the singularities (these are poses in which the manipulator is uncontrollable), answers to the question if non-singular assembly mode change is possible, if a change between operation modes can occur and how the design parameters influence these kinematic properties. New and interesting features of such manipulators are the asymptotic behavior of singularities and higher singularities. The first feature is linked to topological properties of the manipulator and the second one can be investigated in detection of singularities of the singularity locus. The main tools used in this project rely on efficient algebraic and geometric techniques: Studys kinematic mapping, quaternion conjugation, screw theory, line geometry and primary decomposition. Major results This project has brought significant advances in the kinematic analysis of parallel manipulators with less than six degrees of freedom. Thanks to the implementation of efficient tools coming from the complementary expertise of the consortium, several difficult issues have been solved: determination of the constraint equation, characterization and enumeration of the operation modes, singularity analysis, influence of the design parameters on the number and type of operation modes, assembly modes and singularities. Several families of parallel manipulators with various complexities have been studied, which include much more architecture types than initially expected. As an application of this work, a reconfigurable parallel manipulator has been designed.